Hi, I have two of something that looks like this plant. It has little flowers at the end of these succulent type leaves which are pink (one plant) or white (other plant). I have mine outside planted on a tree fern base. Maybe it is a Christmas cactus, but, you know, the leaves are not all that serrated and while the flowers are nice, they are not as showy as some that I have seen on the web. Do you feel this is the same as your one ? I don't think it could make your parents cough..... I would not chuck the plant out, rather your parents could visit their Doctors if they were concerned about their cough.

It's absolutely within the group of flowering succulents, Zygocactus, Schlumbergera and Hatiora are the Thanksgiving/Christmas/Easter cacti. This is less common, with smooth leaf edges, than the widely sold 'Christmas' hybrid ones with soft pointy teeth on the edges. This may bloom in spring. If so , see if it has these 'daisy' like blooms instead of the weird shaped Christmas cactus bloomhttps://en.wikipedia.org/wiki/Hatiora_gaertneri
These plants are very, very common and popular. How would it be causing a cough? But if they really think so, get rid of it, or at least close it in a plastic bag for a while. Won't hurt the plant unless the bag gets really wet with condensation.

Many people are overly concerned about plant toxicity, especially when the plant seems 'unusual' to them in any way.

I have never heard of anyone concerned about plant toxicity before. I have never heard of the idea of a pot plant making a person sick. We (humans) must be losing our connection with outdoors and nature is all I can say.

Plants can be toxic or just irritating in several ways, but I just don't see this one as causing coughing.
Plants with little hairs such as Platanus can be a problem, usually with skin irritation, but I could see it being a respiratory issue if you got these little hairs in your mouth or nose.
Plants with rich fragrance (mint family, jasmines, Eucalyptus and many others) can be a problem. I dealt with a customer a few months ago who said, "No Gardenia, no Jasmine, no strong fragrances... we are allergic". Whether this was a true allergy in the medical sense, I have no idea. I simply avoided using any fragrant plants for them.
Pollen is certainly an allergen for many people, but I think the reaction is not usually coughing.

As I suggested above: IF this plant is the problem, it is not likely the plant itself, but perhaps something growing (fungus) in the soil.
Easiest test: Remove the plant from the house. Clean the area (shelf, walls) around where the plant was. Give the cough some time to clear up. This still does not prove the plant (or soil) caused it. The cough might have cleared up even if the plant stayed there.

I like Hillegonda's suggestion of Jan 7th:
If you are worried about this cough, then see your doctor.

My parents have this plant already 40 years in their bedroom. When I was a kid my grandmother was a great fan of zygocactus and Hibiscus chinensis. I remember their room filled with the reds and pink of numerous varieties of both species. Neither my parents nor grand parents ever had health problems due to it. In general, indoor plants are selected because they are not harmful. Dieffenbachia is one of the few poisonous plants that are still favorite in the livingroom yet who would fancy to eat them. With garden plants that is different. I know a few garden plants that could kill and are very attractive to children but they are beautiful and the traders keep selling them.